In this paper, we study the problem of efficiently reducing geometric shapes into other such shapes in a distributed setting through size-changing operations. We develop distributed algorithms using the \emph{reconfigurable circuit} model to enable fast node-to-node communication. Our study considers two graph update models: the \emph{connectivity} model and the \emph{adjacency} model. Let $n$ denote the number of nodes and $k$ the number of turning points in the initial shape. In the connectivity model, we show that the system of nodes can reduce itself from any tree to a single node using only \emph{shrinking} operations in $O(k \log n)$ rounds w.h.p. and any tree to its minimal (incompressible) form in $O(\log n)$ rounds with additional knowledge or $O(k \log n)$ without, w.h.p. We also give an algorithm to transform any tree to any topologically equivalent tree in $O(k \log n+\log^2 n)$ rounds w.h.p. if both \emph{shrinking} and \emph{growth} operations are available to the nodes. On the negative side, we show that one cannot hope for $o(\log^2 n)$-round transformations for all shapes of $O(\log n)$ turning points: for all reasonable values of $k$, there exists a pair of geometrically equivalent paths of $k$ turning points each, such that $\Omega(k\log n)$ rounds are required to reduce one to the other. In the adjacency model, we show that the system can reduce itself from any connected shape to a single node using only shrinking in $O(\log n)$ rounds w.h.p.

翻译：本文研究在分布式环境下通过尺寸变化操作高效地将几何形状归约为其他形状的问题。我们利用\emph{可重构电路}模型开发分布式算法以实现快速的节点间通信。研究考虑两种图更新模型：\emph{连通性}模型与\emph{邻接}模型。令$n$表示节点数，$k$表示初始形状的转折点数。在连通性模型中，我们证明节点系统仅通过\emph{收缩}操作即可在$O(k \log n)$轮内以高概率将任意树归约为单节点，并在具备额外知识时以$O(\log n)$轮（或无额外知识时以$O(k \log n)$轮）将任意树归约为其最小（不可压缩）形式。若节点同时具备\emph{收缩}与\emph{生长}操作能力，我们给出以$O(k \log n+\log^2 n)$轮将任意树变换为任意拓扑等价树的算法。在负面结果方面，我们证明对于所有具有$O(\log n)$个转折点的形状，不能期望实现$o(\log^2 n)$轮的变换：对于所有合理的$k$值，存在一对各含$k$个转折点的几何等价路径，使得将其中一条归约为另一条需要$\Omega(k\log n)$轮。在邻接模型中，我们证明系统仅通过收缩操作即可在$O(\log n)$轮内以高概率将任意连通形状归约为单节点。